Question: The graph of a sinusoidal function intersects its midline at $(0,-6)$ and then has a minimum point at $(2.5,-9)$. Write the formula of the function, where $x$ is entered in radians. $f(x)=$
Solution: The strategy First, let's use the given information to determine the function's amplitude, midline, and period. Then, we should determine whether to use a sine or a cosine function, based on the point where $x=0$. Finally, we should determine the parameters of the function's formula by considering all the above. Determining the amplitude, midline, and period The midline intersection is at $y={-6}$, so this is the midline. The minimum point is $3$ units below the midline, so the amplitude is ${3}$. The minimum point is $2.5$ units to the right of the midline intersection, so the period is $4\cdot 2.5={10}$. [Why did we multiply by 4?] Determining the type of function to use Since the graph intersects its midline at $x=0$, we should use the sine function and not the cosine function. This means there's no horizontal shift, so the function is of the form $a\sin(bx)+d$. [How do we know that?] Determining the parameters in $a\sin(bx)+d$ Since the midline intersection at $x=0$ is followed by a minimum point, we know that $a<0$. [How do we know that?] The amplitude is ${3}$, so $|a|={3}$. Since $a<0$, we can conclude that $a=-3$. The midline is $y={-6}$, so $d=-6$. The period is ${10}$, so $b=\dfrac{2\pi}{{10}}=\dfrac{\pi}{5}$. The answer $f(x)=-3\sin\left(\dfrac{\pi}{5}x\right)-6$